We say that two sets S, T ⊂ {1, 2, . . . , n} are chord separated if there does not exist a cyclically ordered quadruple a, b, c, d of integers satisfying a, c ∈ S \ T and b, d ∈ T \ S. This is a weaker version of Leclerc and Zelevinsky's weak separation. We show that every maximal by inclusion collection of pairwise chord separated sets is also maximal by size. Moreover, we prove that such collections are precisely vertex label collections of fine zonotopal tilings of the three-dimensional cyclic zonotope. In our construction, Postnikov's plabic graphs and square moves appear naturally as horizontal sections of zonotopal tilings and their mutations, respectively.

We show that the coordinate ring of an open positroid variety coincides with the cluster algebra associated to a Postnikov diagram. This confirms conjectures of Postnikov, Muller-Speyer, and Leclerc, and generalizes results of Scott and Serhiyenko-Sherman-Bennett-Williams. • Π v,w ] with a basis of theta functions with positive structure constants. Additionally, the results of [LS16] imply that H * ( • Π v,w , C) satisfies the

Jim Propp recently introduced a variant of chip-firing on a line where the chips are given distinct integer labels. Hopkins, McConville, and Propp showed that this process is confluent from some (but not all) initial configurations of chips. We recast their set-up in terms of root systems: labeled chip-firing can be seen as a root-firing process which allows the moves λ− →λ+α for α ∈ Φ + whenever λ, α ∨ = 0, where Φ + is the set of positive roots of a root system of Type A and λ is a weight of this root system. We are thus motivated to study the exact same root-firing process for an arbitrary root system. Actually, this central root-firing process is the subject of a sequel to this paper. In the present paper, we instead study the interval root-firing processes determined by λ, for any k ≥ 0. We prove that these interval-firing processes are always confluent, from any initial weight. We also show that there is a natural way to consistently label the stable points of these interval-firing processes across all values of k so that the number of weights with given stabilization is a polynomial in k. We conjecture that these Ehrhart-like polynomials have nonnegative integer coefficients.

No abstract

Zamolodchikov periodicity is a property of certain discrete dynamical systems associated with quivers. It has been shown by Keller to hold for quivers obtained as products of two Dynkin diagrams. We prove that the quivers exhibiting Zamolodchikov periodicity are in bijection with pairs of commuting Cartan matrices of finite type. Such pairs were classified by Stembridge in his study of W -graphs. The classification includes products of Dynkin diagrams along with four other infinite families, and eight exceptional cases. We provide a proof of Zamolodchikov periodicity for all four remaining infinite families, and verify the exceptional cases using a computer program.